Renormalized tropical field theory

We introduce tropical scalar field theory as a model for renormalizable quantum field theory, and examine in detail the case of quartic self-interaction and internal $O(N)$ symmetry. This model arises in a formally zero-dimensional limit of critical long-range models, but nevertheless its Feynman integrals exhibit strong numerical correlations with the ordinary 4-dimensional theory. The tropical theory retains the full complexity of renormalization with nested and overlapping vertex subdivergences and infinitely many primitive graphs. We compute the perturbation series of the tropical renormalization group functions exactly to 400 loops and study their asymptotic growth. In the minimal subtraction scheme, we find only an arithmetic sequence of singularities on the negative real axis in the Borel plane. These singularities are confluent and imply that the large-order perturbative asymptotics contain logarithmic and fractional power corrections. The absence of any further singularities suggests these series are Borel summable. In contrast, in a kinematic subtraction scheme, the singularity structure on the negative axis changes, and we find additional singularities on the positive real axis.